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PARTLY ORDERED SETS 1 Set theoretical notations
Partly ordered sets r
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a r\b a w b arbitrary elements atoms axioms Birkhoff lattice Boolean algebra Boolean ring called closure operation complemented lattice complete lattice congruence relation Consider Corollary cp(x decomposition defined definition denote the set direct union distributive lattice dual equivalence relation example exists finite number Galois connection greatest element Hence holds homomorphism ideal implies inequality infimum infinite irreducible lattice bounded lattice of finite lattice satisfying lattice theory least element Lemma linear subspace locally finite length lower covering condition mapping Math maximal chain maximal element maximum condition modular lattice order isomorphic order preserving pair of elements partly ordered set Proof proposition proved relatively complemented relatively complemented lattice satisfies the maximum Section semimodular lattice Show statement subalgebra subchain subdirect union subgroups sublattice subset lattice Theorem triplet of elements upper bound x w y